An Unconventional Quadrature Method for Logarithmic-Kernel Integral Equations Equations on Closed Curves

A new, fully discrete method is proposed for the logarithmic-kernel integral equation of the first kind on a smooth closed curve. The method uses two levels of numerical quadrature: a trapezoidal rule for the integral containing the logarithmic singularity; and a special quadrature rule for the outer integral, which compensates, in part, for the errors in the first integral. A convergence and stability analysis is given, and the predicted orders of convergence verified in a numerical example. A numerical experiment suggests that the method can be useful even for a curve with corners.